9 THE PLANCHEREL-GODEMENT THEOREM 379
an isomorphism T0 of the prehilbert space A/ng onto its image in #2(8^)
such that T0 - 7ig(x) = x, it is enough to show that this image is dense in
«?(Sff) n ^(Sg,mg)y with respect to the topology of&l(Sg,mg). Given e > 0
and any function F e JTC(S^)3 it is sufficient to prove the existence of x e A
such that
I
Now, we have seen earlier that there exists a function ti = %xixi (xteA)
i
which does not vanish on Supp(F), so that we may write F = Gfi, where
G 6 «?TC(S0). By virtue of (ii), there exists y e A such that
for all x e Sg > and it follows that
(W) - fl(*)Kx)l2 dmg(x) £ (8/N2(fl))2 f |
Finally, for x and y in A, we have by definition
• 71,60) = T0 • ng(xy) = *j> = M(x) • (T0 - ng
and by extending T0 by continuity to an isomorphism Tof Hg onto Lc(S^, mg)
we have TUg(x) = M(x)T. The equality of the norm of 1 • lHg + Ug(x) in
^(VLg) and the norm of 1 + x in ^C(S£) is a consequence of the Gelfand-
Neumark theorem (15.4.14).
(15.9.2.22) Uniqueness of Sg and mg.
It is immediately checked that the hermitian forrn^' satisfies (15.6.3) and
(15.7.3), hence is a bitrace on A. Since the functions x are bounded, we have
g'(xy, xy) = [ \x(X)\2\P(x)\2 dmtf) £ \\x\\2g'(y, y\
J
so that g' satisfies the condition (U). To prove that g' satisfies (N), let S'
denote the closure of S in H(A) u {0}, so that S' = S if S is compact, and
S' = S u {0} otherwise. By virtue of (13.11.6), it is enough to show that the
functions A + xj>, where A G C and x, y e A, form a total set in the space
<^C(S'). The proof is the same as in (15.9.2.11), using the Stone-Weierstrass
theorem and the fact that by definition the functions x separate the points of
S u {0}. The ideal ng, is the set of all x e A for which x is /w-negligible; but
since the support of m is S, and x is continuous, x can be m-negligible only if
x = 0, and therefore ng>(x) H-» x is an isomorphism of A/% onto the algebra of
the functions x such that (ng,(x) \ ng,(yj) = \x(y)$(l) dm(%). The argument of
(15.9.2.21) then shows that the set of functions x is dense in JSfc(S, m), and and therefore definesions xnxn therefore converge uniformly on S^ to G, and hence,